Yuppietown has two food stores, La Boulangerie, which
sells bread, and La Fromagerie, which sells cheese. It costs $1 to
make a loaf of bread and $2 to make a pound of cheese. If La
Boulangerie’s price is P1 dollars per loaf of bread and La
Fromagerie’s price is P2 dollars per pound of cheese, their respective
weekly sales, Q1 thousand loaves of bread and Q2 thousand
pounds of cheese, are given by the following equations:
(a) For each store, write its profit as a function of P1
and P2 (in the exercises that follow, we will call this “the profit function”
for brevity). Then find their respective best-response rules. Graph the
best-response curves, and find the Nash equilibrium prices in this game.
(b) Suppose that the two stores collude and set prices
jointly to maximize the sum of their profits. Find the joint profit-maximizing
prices for the stores.
(c) Provide a short intuitive explanation for the
differences between the Nash equilibrium prices and those that maximize joint
profit. Why is joint profit maximization not a Nash equilibrium?
(d) In this problem, bread and cheese are mutual
complements. They are often consumed together; that is why a drop in the price
of one increases the sales of the other. The products in our bistro example in
Section 1.A are substitutes for each other. How does this difference explain
the differences among your findings for the best-response rules, the Nash
equilibrium prices, and the joint profit-maximizing prices in this question,
and the corresponding entities in the bistro example in the text?